Autonomous random samples of sizesN 1 andN 2 from multivariate common populationsN p (θ1,∑1) andN p (θ2,∑2) are thought of. lower than the null hypothesisH zero: θ1=θ2, a unmarried θ is generated from aN p(μ, Σ) past distribution, whereas underH 1: θ1≠θ2 skill are generated from the exchangeable priorN p(μ,σ). In either situations Σ can be assumed to have a obscure previous distribution. For an easy covariance constitution, the Bayes factorB and minimal Bayes think about favour of the null hypotheses is derived. The Bayes chance for every speculation is derived and a method is mentioned for utilizing the Bayes issue and Bayes hazards to check the speculation.

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Additional info for A Bayesian Approach to the Multivariate Behrens-Fisher Problem Under the Assumption of Proportional Covariance Matrices

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9. Results of calculations, using two-dimensional theory, of thrust coefficient CT and propulsive efficiency 17, as functions of frequency parameter uc/U and of proportional feathering 8, for lunate-tail motion in the carangiform mode. gradient at right angles to the fin equals the rate at which the fin's movement is causing the local water momentum in that direction to change. Figure 9 shows results of some calculations of this kind. The tail is supposed to oscillate laterally with maximum velocity W and maximum angle of attack (reached at the same phase) BW/U.

When this lateral momentum gets left behind (due to the fish's motion at speed U) in the form of a vortex wake shed behind the trailing edge, the force of reaction on the fish is a rate of change of water momentum proportional to Uw. Against this force, the trailing edge moving at velocity W does work at a rate proportional to UwW. However, part of this work is producing the 'wasted' energy of the vortex wake, which is the energy of motion at velocity HYDROMECHANICS OF AQUATIC ANIMAL PROPULSION 21 w shed at velocity U and so is proportionalfto [7(jw2).

4 mm. Positions of maximum curvature are marked with dots and crosses to show backward movement of waves. 6 V for anguilliform propulsion). This formula is explained in Figure 5, but we may note that w must obviously get smaller and smaller as U tends towards V, because in the limiting case when U=V the fish would be slipping through the water without giving it any lateral displacement at all. The undulating fish as it moves through the water has the effect on any particular vertical slice of water (at right angles to the direction of motion) that it gets a part of the slice near the fish into lateral motion, which by the time the fish's trailing edge reaches the slice is motion with velocity w.